Solving System of Equations Using Substitution

Given the equations $y = 5 - 2x$ and $4x + 3y = 13$, solve for $x$ and $y$ using substitution.

In solving systems of equations through substitution, it is essential to focus on isolating one variable in one of the given equations. In this particular example, we have two equations: $y = 5 - 2x$ and $4x + 3y = 13$. The principle behind substitution is to express one variable in terms of the other and substitute this expression into the second equation. This helps in reducing the system of equations to a single equation with one variable, which is easier to solve. Once you solve for one variable, you can backtrack to find the value of the second variable.

Substitution is one of the cornerstone techniques for solving systems of equations and is particularly useful when one of the equations is already or can easily be solved for a variable. This method is advantageous when dealing with straightforward linear equations, where the substitution process is simple and leads directly towards finding a solution. It is an excellent technique to ensure that students grasp the fundamental understanding of how variables interact within a system and how changing one affects the other.

When practicing this method, it is important to verify the solution by substituting the found values back into the original equations. This verification step ensures that the solution satisfies both equations, which is a critical part of confirming the correctness of your result. As students progress, they can apply this method to more complex systems or explore alternative methods such as elimination or matrix operations.